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Why rooting fails

Michael Creutz

TL;DR

The paper investigates why rooting in staggered fermion lattice QCD yields unphysical nonperturbative predictions. It shows that averaging over tastes with different chiralities prevents the correct implementation of the $2N_f$-fermion 't Hooft vertex, thereby misrepresenting instanton-induced effects and mass renormalization. While rooting can be perturbatively justified and may perform well for some observables, it cannot be exact in the continuum limit due to intrinsic chiral and topological mismatches, especially in singlet channels. The findings have significant implications for the reliability of nonperturbative results obtained with rooted staggered fermions and motivate exploring alternatives such as square-root determinants, counter-terms, or chiral ghost formulations.

Abstract

I explore the origins of the unphysical predictions from rooted staggered fermion algorithms. Before rooting, the exact chiral symmetry of staggered fermions is a flavored symmetry among the four "tastes." The rooting procedure averages over tastes of different chiralities. This averaging forbids the appearance of the correct 't Hooft vertex for the target theory.

Why rooting fails

TL;DR

The paper investigates why rooting in staggered fermion lattice QCD yields unphysical nonperturbative predictions. It shows that averaging over tastes with different chiralities prevents the correct implementation of the -fermion 't Hooft vertex, thereby misrepresenting instanton-induced effects and mass renormalization. While rooting can be perturbatively justified and may perform well for some observables, it cannot be exact in the continuum limit due to intrinsic chiral and topological mismatches, especially in singlet channels. The findings have significant implications for the reliability of nonperturbative results obtained with rooted staggered fermions and motivate exploring alternatives such as square-root determinants, counter-terms, or chiral ghost formulations.

Abstract

I explore the origins of the unphysical predictions from rooted staggered fermion algorithms. Before rooting, the exact chiral symmetry of staggered fermions is a flavored symmetry among the four "tastes." The rooting procedure averages over tastes of different chiralities. This averaging forbids the appearance of the correct 't Hooft vertex for the target theory.

Paper Structure

This paper contains 7 sections, 10 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Naive justifications for rooting
  3. Staggered review
  4. Where things go awry
  5. The 't Hooft vertex
  6. Questions
  7. Conclusion

Figures (5)

  • Figure 1: When a fermion circumnavigates a loop in the naive formulation, it picks up a factor that always involves an even power of any particular gamma matrix.
  • Figure 2: In the overlap formulation a single exact zero eigenmode is possible. In transiting from winding number zero to one, a pair of complex eigenvalues disappear and are replaced with the exact zero mode and a compensating mode on the opposite side of the overlap circle.
  • Figure 3: In transiting between different winding number sectors, the clustering of the staggered Dirac eigenvalues into taste quartets must break down.
  • Figure 4: With $N_f$ flavors the 't Hooft vertex is a $2N_f$ fermion interaction where each flavor flips its spin.
  • Figure 5: In calculating the instanton/anti-instanton interaction, a contribution will arise from the exchange of all tastes. This introduces an unphysical singularity in the rooted theory.